On the Consistency of Ordinal Regression Methods

COUV_CAHIER_EGND_4by Fabian Pedregosa, Francis Bach & Alexandre Gramfort

Many of the ordinal regression models that have been proposed in the literature can be seen as methods that minimize a convex surrogate of the zero-one, absolute, or squared loss functions. A key property that allows to study the statistical implications of such approximations is that of Fisher consistency. In this paper we will characterize the Fisher consistency of a rich family of surrogate loss functions used in the context of ordinal regression, including support vector ordinal regression, ORBoosting and least absolute deviation. We will see that, for a family of surrogate loss functions that subsumes support vector ordinal regression and ORBoosting, consistency can be fully characterized by the derivative of a real-valued function at zero, as happens for convex margin-based surrogates in binary classication. We also derive excess risk bounds for a surrogate of the absolute error that generalize existing risk bounds for binary classication. Finally, our analysis suggests a novel surrogate of the squared error loss. To prove the empirical performance of such surrogate, we benchmarked it in terms of cross-validation error on 9 different datasets, where it outperforms competing approaches on 7 out of 9 datasets.

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